# A simple example for the application of the Hartree-Fock method?

I'm currently reading a book about electronic structure, and I'm really interested in Hartree-Fock method. But I can only have an empirical understanding from the book. So I would like to ask for one or perhaps two simple, basic mathematical examples that I can work with to gain a better understanding of the method.

• This post (v4) seems like a list question. Aug 4, 2017 at 5:17
• @Qmechanic its a good question and I'd enjoy reading an answer and trying out the problem myself! Hopefully it's been sufficiently "de-listed" (de-listicized?)
– uhoh
Aug 4, 2017 at 6:53

I would suggest that you try to write a small code that can perform a Hartree-Fock calculation.

1. Specify molecule and basis set.

2. Get the integrals. The integrals can be hard to calculate on your own so I would suggest to obtain those from different sources, see Programming Project #3: The Hartree-Fock Self-Consistent Field Method

3. Make a diagonalization of the overlap matrix, $S$, to get the transformation matrix, $S^{-1/2}$. $$S^{-1/2}=L_s\lambda_s^{-1/2}L_s^T$$ Here $L_s$ is the eigenvectors of the overlap matrix, and $\lambda_s$ is the diagonalized overlap matrix.

4. Make an initial guess for the density matrix, $D$. This can be done using the core Hamiltonian $H_{core}$ as an inital guess for the Fock matrix, $F$. The core Hamiltonian is given as the sum of the kinetic energy and electron nuclei coulomb interaction integrals. Now orthonormal Fock matrix can be constructed: $$F'_0 = S^{-1/2,T}H_{\mathrm{core}}S^{-1/2}$$ This can now be diagonalized to obtain orbtial coefficients, $C'$: $$F'_0C'_0=C'_0\epsilon_0$$ These coefficient can be back transformed $C=S^{-1/2}C'$ and used to construct the density matrix: $$D_{ij,0} = \sum_{ij}^{occ}C_{i,0}C_{j,0}$$ The inital energy can also be found as $E_{elec,0} = \sum_{ij}^{occ}D_{ij,0}(H_{ij,\mathrm{core}}+H_{ij,\mathrm{core}})$.

5. Build new Fock matrix. $$F_{ij} = H_{ij,\mathrm{core}} + \sum_{kl}^{occ}D_{kl,0}(2(ij|kl)-(ik|jl))$$

6. Build new density matrix, in the same way as the inital density matrix was build. $$F' = S^{-1/2,T}FS^{-1/2}$$ $$F'C'=C'\epsilon$$ $$D_{ij} = \sum_{ij}^{occ}C_{i}C_{j}$$ The energy can now be found as $E_{elec} = \sum_{ij}^{occ}D_{ij}(F_{ij}+H_{ij,\mathrm{core}})$. Return to step 5 until convergence is reached.

• Holy granola what a transcription, very nice. +n! :)